Transcript - Harish-Chandra Research Institute

Quantum Simulations by NMR:
Recent Developments
Anil Kumar1, K. Rama Koteswara Rao1, T.S. Mahesh2 and
V.S.Manu1
Indian Institute of Science, Bangalore-560012
Indian Institute of Science Education and Research, Pune-411 008
QIPA-December 2013, HRI-Allahabad
In 1982, Feynman suggested that it might be possible to simulate
the evolution of quantum systems efficiently, using a quantum
simulator
Recent Developments
Simulating a Molecule: Using Aspuru-Guzik Adiabatic Algorithm
(i) J.Du, et. al,
Phys. Rev. letters 104, 030502 (2010).
Used a 2-qubit NMR System ( 13CHCl3 ) to calculated the
ground state energy of Hydrogen Molecule up to 45 bit accuracy.
(ii) Lanyon et. al,
Nature Chemistry 2, 106 (2010).
Used Photonic system to calculate the energies of the ground
and a few excited states up to 20 bit precision.
Experimental Techniques for Quantum Computation:
1. Trapped Ions
Ion Trap:
2. Polarized Photons 3. Cavity Quantum
Electrodynamics (QED)
Lasers
1kV
Linear Paul-Trap
Laser
Cooled
Ions
T ~ mK
~
16 MHz
1kV
http://heart-c704.uibk.ac.at/linear_paul_trap.html
4. Quantum Dots
5. Cold Atoms
7. Josephson junction qubits
8. Fullerence based ESR quantum computer
6. NMR
Nuclear Magnetic Resonance (NMR)
1. Nuclear spins have small magnetic
moments and behave as tiny quantum
magnets.
2. When placed in a magnetic field
(B0), spin ½ nuclei orient either
along the field (|0 state) or opposite
to the field (|1 state) .
B1
0
3. A transverse radio-frequency field (B1) tuned at the Larmor frequency of
spins can cause transition from |0 to |1 (NOT Gate by a 1800 pulse).
Or put them in coherent superposition (Hadamard Gate by a 900 pulse).
Single qubit gates.
4. Spins are coupled to other spins by indirect spin-spin (J) coupling, and
controlled (C-NOT) operations can be performed using J-coupling.
Multi-qubit gates
NUCLEAR SPINS ARE QUBITS
DSX 300
7.0 Tesla
AV 700
16.5 Tesla
NMR Research Centre, IISc
Field/
Frequency
stability
= 1:10 9
1 PPB
DRX 500
11.7 Tesla
AV 500
11.7 Tesla
AMX 400
9.4 Tesla
NMR Facility in IISc
800 MHz NMR spectrometer equipped with cryogenic probe
Installed in April 2013
18.8 Tesla
Why NMR?
> A major requirement of a quantum computer is that the
coherence should last long.
> Nuclear spins in liquids retain coherence ~ 100’s millisec
and their longitudinal state for several seconds.
> A system of N coupled spins (each spin 1/2) form an N
qubit Quantum Computer.
> Unitary Transform can be applied using R.F. Pulses and
J-evolution and various logical operations and quantum
algorithms can be implemented.
Addressability in NMR
NMR sample has ~ 1018 spins.
Do we have 1018 qubits?
No - because, all the spins can’t be
individually addressed.
Progress so far
Spins having different Larmor frequencies can be addressed in
the frequency domain resulting-in as many “qubits” as Larmor
frequencies, each having 1018 spins. (ensemble computing)
One needs resolved couplings between the spins
in order to encode information as qubits.
NMR Hamiltonian
H = HZeeman + HJ-coupling
Two Spin System (AM)
=  wi Izi +  Jij Ii  Ij
i
bb = 11
i<j
Weak coupling Approximation
wi - wj>> Jij
H =  wi Izi +  Jij Izi Izj
i
i<j
A2= 1M
M2= 1A
ab = 01
ba = 10
M1= 0A
A1= 0M
aa = 00
Spin States are eigenstates
A2 A1
Under this approximation spins
having same Larmor Frequency
can be treated as one Qubit
wA
M2 M1
wM
NMR Qubits
An example of a three qubit system.
13CHFBr
2
13C
1H
= 500 MHz
13C
= 125 MHz
JCH = 225 Hz
JCF = -311 Hz
JHF = 50 Hz
19F
= 470 MHz
Br (spin 3/2) is a quadrupolar nucleus, is decoupled
from the rest of the spin system and can be ignored.
Homo-nuclear spins having different Chemical shifts
(Larmor frequencies) also form multi-qubit systems
3 Qubits
2 Qubits
1 Qubit
111
CHCl3
11
1
0
10
011 110
01
00
010
001 100
000
101
Pure State
Tr(ρ ) = Tr ( ρ2 ) = 1
For a diagonal density matrix, this condition requires that all energy
levels except one have zero populations
Such a state is difficult to create in room temperature liquid-state
NMR
Pseudo-Pure State
Under High Temperature Approximation
0
 11
1
 01
ρ = 1/N ( α1 - Δρ )
2
 00
Here α = 105 and U 1 U-1 = 1
We create a state in which all levels
except one have EQUAL populations.
Such a state mimics a pure state.
1
 10
0
 11
0
 01
4
 00
0
 10
Achievements of NMR - QIP

1. Preparation of
Pseudo-Pure States

2. Quantum Logic Gates


3. Deutsch-Jozsa Algorithm



5. Hogg’s algorithm


4. Grover’s Algorithm




10. Quantum State Tomography
11. Geometric Phase in QC
12. Adiabatic Algorithms
13. Bell-State discrimination
14. Error correction
15. Teleportation
6. Berstein-Vazirani parity algorithm
16. Quantum Simulation
7. Quantum Games
17. Quantum Cloning
8. Creation of EPR and GHZ states
18. Shor’s Algorithm
9. Entanglement transfer


19. No-Hiding Theorem
 Also performed in our Lab.
Maximum number of qubits achieved in our lab: 8
In other labs.: 12 qubits;
Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, 170501 (2006).
Recent Publications from our Laboratory
1.
Experimental Test of Quantum of No-Hiding theorem.
Jharana Rani Samal, Arun K. Pati and Anil Kumar,
Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)
2. Use of Nearest Neighbor Heisenberg XY interaction for creation
of entanglement on end qubits in a linear chain of 3-qubit
system.
K. Rama Koteswara Rao and Anil Kumar
IJQI, 10, 1250039 (2012).
3. Multi-Partite Quantum Correlations Revel Frustration in a
Quantum Ising Spin System.
K. Rama Koteswara Rao, Hemant Katiyar, T.S. Mahesh, Aditi Sen (De), Ujjwal Sen and
Anil Kumar:
Phys. Rev A 88, 022312 (2013).
4. Use of Genetic Algorithm for QIP.
V.S. Manu and Anil Kumar,
Phys. Rev. A 86, 022324 (2012)
No-Hiding Theorem
S.L. Braunstein & A.K. Pati, Phys.Rev.Lett. 98, 080502 (2007).
Any physical process that bleaches out the original information
is called “Hiding”. If we start with a “pure state”, this bleaching
process will yield a “mixed state” and hence the bleaching
process is “Non-Unitary”. However, in an enlarged Hilbert
space, this process can be represented as a “unitary”.
The No-Hiding Theorem demonstrates that the initial pure
state, after the bleaching process, resides in the ancilla qubits
from which, under local unitary operations, is completely
transformed to one of the ancilla qubits.
Quantum Circuit for Test of No-Hiding Theorem using State
Randomization (operator U).
H represents Hadamard Gate and dot and circle represent
CNOT gates.
After randomization the state |ψ> is transferred to the second
Ancilla qubit proving the No-Hiding Theorem.
(S.L. Braunstein, A.K. Pati, PRL 98, 080502 (2007).
NMR Pulse sequence for the Proof of No-Hiding Theorem
The initial State ψ is
prepared for different
values of θ and φ
Jharana Rani et al
Pulse Sequence for U obtained
by using the method out lined in
A Ajoy, RK Rao, A Kumar, and P
Rungta, Phys. Rev. A 85,
030303(R) (2012)
Experimental Result for the No-Hiding Theorem.
The state ψ is completely transferred from first qubit to the third qubit
Input State
s
Output State
s
S = Integral of real part of the signal for each spin
325 experiments have been performed by varying θ and φ in steps of 15o
All Experiments were carried out by Jharana (Dedicated to her memory)
Jharana Rani Samal, Arun K. Pati and Anil Kumar,
Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)
Use of nearest neighbor
Heisenberg-XY interaction.
Until recently we had been looking for qubit systems, in which
all qubits are coupled to each other with unequal couplings, so
that all transitions are resolved and we have a complete access
to the full Hilbert space.
However it is clear that such systems are not scalable,
since remote spins will not be coupled.
Solution
Use Nearest Neighbor Interactions
In a first study towards this goal,
we took a linear chain of 3-qubits
And
Using only the Nearest-Neighbor
Heisenberg XY Interaction,
Entangle end qubits as well as Entangle all the 3 qubits
Heisenberg XY interaction is normally not present in
liquid-state NMR: We have only ZZ interaction.
We create the XY interaction by transforming the ZZ
interaction by the use of RF pulses.
Koteswara Rao et al, IJQI, 10 (4) 1250039 (2012)
Nearest-Neighbour Heisenberg XY Interaction
H XY =
N -1
1
2
x x
y y
J
(


+

 i i i+1 i  i+1 )
i =1
N -1

=  J i  i+ i-+1 +  i- i++1
i =1

Consider a linear Chain of 3 spins with equal couplings
J
J
1
2
3

H XY = 12 J  1x x2 +  1y y2 +  x2 x3 +  y2 y3

σjx/y are the Pauli spin matrices and J is the coupling constant between two spins.
U (t ) = e
Divide the HXY into two commuting parts
-iH XY t
  +   ,   +    = 0
- 2i Jt  1x x2 + y2 3y  - 2i Jt  1y y2 + x2 x3 
U (t ) = U A (t ) U B (t ) = e
e
1
x

U A (t ) = e
- Jt  
U B (t ) = e
2
x
2
y
3
y
1
y
2
y
2
x
3
x

+  
- 2i Jt  1x x2 + y2 3y
i
2
1
y
2
y
2
x
3
x
Jingfu Zhang et al., Physical Review A, 72, 012331(2005)
The operator U in Matrix form for the 3-spin system
0
0
0
0
0
0
1


2
2
i
0
cos

sin(
2

)
0
sin

0
0
2


i
0 - i sin(2 )
cos(
2

)
0
sin(2 )
0
0
2
2


0
0
cos2 
0
- i2 sin(2 )
- sin 2 
0
U (t ) = 
0
- sin 2 
- i2 sin(2 )
0
cos2 
0
0


0
0
- i2 sin(2 )
0
cos(2 )
- i2 sin(2 )
0


2
2
i
0
0
0
sin

0
sin(
2

)
cos


2

0
0
0
0
0
0
0
where
0


0

0


0

0


0


0

1
φ = Jt/√2
Jingfu Zhang et al., Physical Review A, 72, 012331(2005)
Two and three qubit Entangling Operators
1 0

1
0
2

i
0

2

0 0
U t  = 
0 - 1

2

0 0

0 0

0 0

0
1


0 0.79

-i
0

3

0
0
U (t ) = 
0 - 0.21


0
0


0
0

0
0
0
-i
2
0
0
0
0
1
2
0
-i
2
0
1
2
0
0
0
-i
2
0
0
0
0
-i
3
1
3
0
-i
2
1
2
0
0
1
2
-i
2
-
0
0
0
0
0
0
0
0
-i
2
0
0
0
0
- 0.21
0
0
-i
3
0
0
0.79
0
-i
3
-i
3
0
0.79
0
0
0
0
- 0.21
0
0
0
0
1
2
0
0
-i
3
-
1
3
-i
3
0
-i
2
1
2
0
0

0

0


0

0


0

0

1 
0


0
0

0
0


- 0.21 0

0
0


-i
0
3


0.79 0

0
1
0
When t =

2 2J
Bell States
 = /4
,
U 101 =
-i
2
 01 + 10 
U 010 =
-i
2
 10
13
1
+ 01 13 0
2
2
W - State
tan-1 ( 2 )
When t =
,
2J
U (t ) 101 =
1
3
tan-1 ( 2 )
=
2
101 -
i
3
011 -
Phase Gate on 2ndspin
0 0 +i 1 1
1
3
101 +
1
3
011 +
1
3
110
i
3
110
Experiment
Equilibrium spectra
Sample
Pulse sequence to prepare PPS (using only the nearest neighbour couplings)
1H
13C
19F
Gz
Tomography
of 010 > PPS
Pulse sequence to simulate XY Hamiltonian
1H
13C
19F
Experimental results
Bell state on end qubits
U 010 =
-i
2
 10 + 01 
13
0
2
Attenuated
Correlation(c)=0.86
U 101 =
-i
2
 01 + 10 
Attenuated
Correlation(c)=0.88
K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).
13
12
Experimental results
W-state
1
3
101 +
1
3
011 +
1
3
110
Attenuated
Correlation(c)=0.89
GHZ-state ( |000> + |111> )/√2
Attenuated
Correlation(c)=0.81
K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).
Quantum simulation of frustrated
Ising spins by NMR
K. Rama Koteswara Rao1, Hemant Katiyar3,
T.S. Mahesh3, Aditi Sen (De)2, Ujjwal Sen2 and
Anil Kumar1:
Phys. Rev A 88 , 022312 (2013).
1
Indian Institute of Science, Bangalore
2 Harish-Chandra Research Institute, Allahabad
3 Indian Institute of Science Education and Research, Pune
A spin system is frustrated when the minimum of the system energy does
not correspond to the minimum of all local interactions. Frustration in
electronic spin systems leads to exotic materials such as spin glasses and
spin ice materials.
3-spin transverse Ising system
If J is negative
Ferromagnetic
The system is non-frustrated
If J is positive
Anti-ferromagnetic
The system is frustrated
Here, we simulate experimentally the ground state of a 3-spin system in both
the frustrated and non-frustrated regimes using NMR.
Experiments at 290 K in a
500 MHz NMR Spectrometer
of IISER-Pune
Diagonal elements are chemical shifts and offdiagonal elements are couplings.
This rotation was realized by a numerically optimized amplitude and phase
modulated radio frequency (RF) pulse using GRadient Ascent Pulse Engineering
(GRAPE) technique1.
1N.
Khaneja and S. J. Glaser et al., J. Magn. Reson. 172, 296 (2005).
Non-frustrated
Frustrated
Multipartite quantum correlations
Entanglement
Score using
deviation
Density matrix
Initial State:
Equal Coherent
Superposition
State. Fidelity = .99
Non-frustrated
regime: Higher
correlations
Frustrated
regime:
Lower
correlations
Ground State
GHZ State (J >> h)
‫׀‬
‫׀‬
( 000> - 111>)/√2
Fidelity = .984
Quantum Discord
Score using full
density matrix
Koteswara Rao et al. Phys. Rev A 88 , 022312 (2013).
Recent Developments in our Laboratory:
(To be published)
1. Simulation of Mirror Inversion of Quantum States in
an XY spin-chain.
2. Quantum Simulation of Dzyaloshinsky-Moriya (DM)
interaction in presence of Heisenberg XY interaction
for study of Entanglement Dynamics
Mirror Inversion of quantum states in an XY spin chain*
1
J1
2
J2
3
N-1
JN-1
N
• The above XY spin chain generates the mirror image of any input state
up to a phase difference.
• Entangled states of multiple qubits can be transferred from one end of the
chain to the other end
*Albanese et al., Phys. Rev. Lett. 93, 230502 (2004)
*P Karbach, and J Stolze et al., Phys. Rev. A 72, 030301(R) (2005)
NMR Hamiltonian of a weakly coupled spin system
Control Hamiltonian
Simulation
In practice
Simulation
1) GRAPE algorithm
2) An algorithm by
A Ajoy et al. Phys. Rev. A 85, 030303(R) (2012)
 Here, we use a combination of these two algorithms to simulate the unitary
evolution of the XY spin chain
4-spin chain
5-spin chain
In the experiments, each of these decomposed operators are simulated
using GRAPE technique
N-spin chains
For odd N
For even N
The number of operators in the decomposition
increases only linearly with the number of spins (N).
Experiment
Molecular structure and Hamiltonian parameters
5-spin system
The dipolar couplings of the spin system
get scaled down by the order parameter
(~ 0.1) of the liquid-crystal medium.
The sample 1-bromo-2,4,5-trifluorobenzene is partially oriented in a liquidcrystal medium MBBA
The Hamiltonian of the spin system in the doubly rotating frame:
Quantum State Transfer: 4-spin pseudo-pure initial states
Diagonal part of the deviation density matrices (traceless)
The x-axis represents the standard computational basis in decimal form
Coherence Transfer: 5-spin initial states
Spectra of Fluorine spins
Proton spins
K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A
Coherence Transfer: 5-spin initial states
Spectra of Fluorine spins
Proton spins
Entanglement Transfer
Initial States
Final States
Reconstructed deviation density matrices (trace less) of spins 1 and 2,
and spins 4 and 5.
K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A
Entanglement Transfer
Initial States
Final States
Reconstructed deviation density matrices (trace less) of spins 1 and 2,
and spins 4 and 5.
Fluorine
transitions
The effective transverse relaxation times (T2*):
40-100ms
Proton
transitions
110-150ms
The Fidelity of the Final state is high. This means that the state is reproduced
with high fidelity. But the attenuated correlations are low. This means the
intensities of the signal in the final state are low compared to initial state. This is
mainly due to decoherence.
The Genetic Algorithm
Charles Darwin 1866
1809-1882
John Holland
Directed search algorithms based on the
mechanics of biological evolution
Developed by John Holland, University of
Michigan (1970’s)
Genetic Algorithm
“Genetic Algorithms are good at taking large, potentially
huge, search spaces and navigating them, looking for
optimal combinations of things, solutions one might not
otherwise find in a lifetime”
Here we apply Genetic Algorithm to Quantum Information Processing
We have used GA for
(1) Quantum Logic Gates (operator optimization)
and
(2) Quantum State preparation (state-to-state optimization)
V.S. Manu et al. Phys. Rev. A 86, 022324 (2012)
Representation Scheme
Representation scheme is the method used for encoding the
solution of the problem to individual genetic evolution. Designing a
good genetic representation is a hard problem in evolutionary
computation. Defining proper representation scheme is the first
step in GA Optimization*.
In our representation scheme we have selected the gene as a
combination of
(i) an array of pulses, which are applied to each channel with
amplitude (θ) and phase (φ),
(ii) An arbitrary delay (d).
It can be shown that the repeated application of above gene forms
the most general pulse sequence in NMR
* Whitley, Stat. Compt. 4, 65 (1994)
The Individual, which represents a valid solution can be
represented as a matrix of size (n+1)x2m. Here ‘m’ is the
number of genes in each individual and ‘n’ is the number
of channels (or spins/qubits).
So the problem is to find an optimized matrix, in which the
optimality condition is imposed by a “Fitness Function”
Fitness function
In operator optimization
GA tries to reach a preferred target Unitary Operator (Utar) from an
initial random guess pulse sequence operator (Upul).
Maximizing the Fitness function
Fpul = Trace (Upul Χ Utar )
In State-to-State optimization
Fpul = Trace { U pul (ρin) Upul (-1) ρtar † }
Single Qubit Rotation in a Two-qubit Homonuclear case
Sequence obtained by
Genetic Algorithm
Non-Selective
(Hard) Pulses
applied at the
centre
π/2
π/2
By changing φ1 and φ2 one
can excite either spin 1 or 2
δ = 500 Hz, J= 3.56 Hz
φ1 = 2π, φ2 = π,
Θ = π/2, φ = π/2
H = 2π δ (I1z – 12z) + 2π J12 (I1zI2z)
Hamiltonian used
Simulated using J = 0
φ1 = π, φ2 = 0
Θ = π/2, φ = π/2
Fidelity for finite J/δ
C-NOT
Sequence obtained by Genetic Algorithm
Equilibrium
0
01
-1
11
1
00
0
10
0
11 -1
0
01
10
1
00
-1
01
0
11
1
00
0
10
1
01
-1
11
0
00
0
10
0
01
-1
11
0
00
1
10
Pseudo Pure State (PPS) creation
Sequence obtained by Genetic Algorithm
All unfilled rectangles represent 900 pulse
The filled rectangle is 1800 pulse.
Phases are given on the top of each pulse.
00
01
10
11
Fidelity w.r.t. to J/δ
00 and 11 PPS are obtained by -/+
on 180 pulse. Other PPS are
obtained by using an additional
180 SQR pulse
Bell state creation:
From Equilibrium (No need of PPS)
Sequence obtained by Genetic Algorithm
0
0
0
All blank pulses are 900 pulses. Filled pulse is a 1800 pulse.
Phases and delays Optimized for best fidelity.
0
The Singlet Bell State
Experimental Fidelity > 99.5 %
Shortest Pulse Sequence for creation
of Bell States directly from Equilibrium
V.S. Manu et al. Phys. Rev. A 86, 022324 (2012)
T1=8.7 s
Ts=11.2 s
(b) Quantum Simulation of Dzyaloshinsky-Moriya (DM)
interaction (HDM) in presence of Heisenberg XY interaction
(HXY) for study of Entanglement Dynamics
Manu et al. (Under Review)
DM Interaction1,2
Anisotropic antisymmetric exchange interaction arising from spin-orbit coupling.
Proposed by Dzyaloshinski to explain the weak ferromagnetism of
antiferromagnetic crystals (Fe2O3, MnCO3).
Quantum simulation of a Hamiltonian H requires unitary operator
decomposition (UOD) of its evolution operator, (U = e-iHt) in terms of
experimentally preferable unitaries.
Using Genetic Algorithm optimization, we numerically evaluate the most generic
UOD for DM interaction in the presence of Heisenberg XY interaction.
1. I. Dzyaloshinsky, J. Phys & Chem of Solids, 4, 241 (1958).
2. T. Moriya, Phys. Rev. Letters, 4, 228 (1960).
The Hamiltonian
Heisenberg XY interaction
DM interaction
Evolution Operator:
Decomposing the U in terms of Single Qubit Rotations (SQR) and ZZevolutions.
SQR by Hard pulse
ZZ evolutions by Delays
Fidelity is defined as.
A. The Decomposition for ϒ = 0 -1:
1
Period of U(ϒ,τ)
This has a maximum value of 12.59.
Optimization is performed for τ -> 0-15, which
includes one complete period.
Fidelity > 99.99 %
Manu et al. (under Review).
ϒ’ = 1/ ϒ
When ϒ > 1 -> ϒ’ < 1
2
Using above decomposition, we studied entanglement preservation in a
two qubit system.
Entanglement Preservation
Hou et al. 1 demonstrated a mechanism for entanglement preservation using H(J,D). They
showed that preservation of initial entanglement is performed by free evolution
interrupted with a certain operator O, which makes the state to go back to its initial state.
concurrence
µi are eigen values of the operator
ρSρ*S, where S= σ1y ⊗ σ2y
Without Operator O
With Operator O
Entanglement (concurrence) oscillates
Entanglement (concurrence) is preserved during
during Evolution.
Evolution. This confirms the Entanglement
preservation method of Hou et al.
(Manu et al. Under Review)
1Hou
et al. Annals of Physics, 327 292 (2012)
Acknowledgements
Former QC- IISc-Associates/Students
Current QC IISc - Students
Prof. Arvind
Prof. Kavita Dorai
Prof. T.S. Mahesh
Dr. Neeraj Sinha
Dr. K.V.R.M.Murali
Dr. Ranabir Das
Dr. Rangeet Bhattacharyya
Mr. K. R. Koteswara Rao
Mr. V.S. Manu
Dr.Arindam Ghosh
Dr. Avik Mitra
Dr. T. Gopinath
Dr. Pranaw Rungta
Dr. Tathagat Tulsi
- IISER Mohali
- IISER Mohali
- IISER Pune
- CBMR Lucknow
- IBM, Bangalore
- NCIF/NIH USA
- IISER Kolkata
- NISER Bhubaneswar
- Philips Bangalore
- Univ. Minnesota
- IISER Mohali
– IIT Bombay
This lecture is dedicated
to the memory of
Ms. Jharana Rani Samal*
(*Deceased, Nov., 12, 2009)
Other IISc Collaborators
Prof. Apoorva Patel
Prof. K.V. Ramanathan
Prof. N. Suryaprakash
Other Collaborators
Prof. Malcolm H. Levitt - UK
Prof. P.Panigrahi
IISER Kolkata
Prof. Arun K. Pati
HRI-Allahabad
Prof. Aditi Sen
HRI-Allahabad
Prof. Ujjwal Sen
HRI-Allahabad
Mr. Ashok Ajoy
BITS-Goa-MIT
Funding: DST/DAE/DBT
Thanks: NMR Research Centres at IISc, Bangalore
and IISER-Pune for spectrometer time
Thank You
Quantum State Transfer
When t =  / 2 J ,  = /2
Two qubit Entangling Operator

When t =
,
2 2J
 = /4
Three qubit Entangling Operator
tan-1 ( 2 )
When t =
,
2J
tan-1 ( 2 )
=
2